Sunday, August 31, 2014

Gillies’ Philosophical Theories of Probability, Chapter 3

Chapter 3 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with Keynes’ logical theory of probability, which was taken up by the Vienna circle and logical positivists like Carnap (Gillies 2000: 25).

Keynes’ logical theory was partly inspired by the lectures of W. E. Johnson at Cambridge university, which were also attended by Harold Jeffreys who later formulated his own logical theory of probability (Gillies 2000: 25).

Keynes had finished the proofs of his Treatise on Probability in 1913, but it was not published until 1921.

Keynes’ views on probability were influenced by the intellectual climate at Cambridge university, particularly the ethical work of G. E. Moore and Bertrand Russell’s logicist work on mathematics (Gillies 2000: 27). Gillies (2000: 27) sees Keynes’ attempts to provide a “logical” foundation for probability, and particularly inductive reasoning, as inspired by Russell and Whitehead’s attempts to found mathematics on logic.

For Keynes, if the evidence h justifies a conclusion a to some degree α, then there is a probability relation of degree α between a and h, so that inductive probability is a degree of partial entailment or degree of rational belief (Gillies 2000: 31)

Keynes thought that we have knowledge of this probability relation by logical intuition or direct acquaintance, but this is a problematic part of Keynes’ theory (Gillies 2000: 31–32).

Gillies (2000: 32–33) argues that underlying Keynes’ idea that probability is objective and known by logical intuition is a problematic concept derived from Platonic ontology: that the probability relation is objective and belongs to some Platonic realm.

Keynes also thought that mathematical probabilities are expressible as numbers on a range in the interval [0, 1]; he thought that not every probability had a numerical value, and also that some probabilities can only be arranged in an ordinal ranking, while others cannot even be compared at all (Gillies 2000: 33–34).

The “principle of indifference,” which was introduced by Bernoulli as the “principle of non-sufficient reason,” is an a priori principle invoked by Keynes to determine when cases are equiprobable (Gillies 2000: 35–36). Gillies (2000: 37–46) sees insuperable difficulties with the a priori principle of indifference, such as the book paradox and wine–water paradox, which Keynes did not really solve.

While the “principle of indifference” might in some cases be a useful heuristic, it cannot be a sound logical principle, and so, since Keynes’ logical theory of probability requires it to be a successful theory, Gillies (2000: 48–49) concludes that Keynes’ overall logical theory is flawed, though important aspects of it may be salvaged, provided that the theory’s Platonist view of how the probability relation is understood and the a priori principle of indifference are abandoned (Gillies 2000: 35, 49; Runde 1994).